Quantum-state-generating apparatus, Bell measurement apparatus, quantum gate apparatus, and method for evaluating fidelity of quantum gate

ABSTRACT

An apparatus for generating a quantum state of a two-qubit system including two qubits, each qubit being represented by a particle which invariably travels through one of two paths, includes a quantum gate composed of an interferometer for implementing an-interaction-free measurement. The apparatus receives two particles having no correlation and generates a Bell state with asymptotic probability 1. A Bell measurement of a state of a two-qubit system is performed by observing a quantum gate composed of the interferometer after the quantum gate has processed the state and selecting the state from the Bell bases. An approximate fidelity of a quantum gate composed of the interferometer is calculated, if an absorption probability with which a first particle absorbs a second particle in the interferometer is less than 1, under the condition that the number of times the second particle hits beam splitters in the interferometer is sufficiently large.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to the field of quantum informationprocessing such as quantum computation, quantum communication, andquantum cryptography, and more specifically relates to an apparatus forgenerating an entangled quantum state of a plurality of qubits(two-state quantum systems), an apparatus for performing Bellmeasurement, which is simultaneous measurement on a two-qubit system, anapparatus for implementing a controlled-NOT gate, which is unitarytransformation of a two-qubit system, and a method for performingapproximate evaluation of the fidelity of a quantum gate composed of aninterferometer which performs an interaction-free measurement (IFM)(hereafter called an IFM interferometer).

2. Description of the Related Art

Since it was found that quantum computation can solve some kinds ofproblems more efficiently than classical computers, quantum computationhas been extensively researched (see references (1) to (4)). Inaddition, as research on quantum information processing, such as quantumteleportation, has become popular, the importance of physical phenomenoncalled quantum entanglement has become more recognized (see references(5) and (6)).

Quantum computation is performed by preparing a plurality of two-statequantum systems called qubits and successively performing unitarytransformation of them and observing the results. Any unitarytransformation applied to the qubits can be decomposed into U(2)transformations applied to a single qubit and controlled-NOT gatesoperating on two qubits (see reference (7)). The controlled-NOT gateproduces quantum entanglement between two qubits, and various methodshave been proposed and experiments have been carried out to realize acontrolled-NOT gate. For example, a method using cavity quantumelectrodynamics (QED) (see references (8) and (9)), a method forimplementing the operation with a certain probability using linearoptical devices (see reference (10)), and a method using asuperconductive Josephson junction (see reference (11)) have beenproposed. However, all of these methods require highly advancedexperimental techniques, and are not expected to be put into practicaluse in the near future. If a controlled-NOT gate were realized, Bellmeasurement, which is simultaneous measurement on a two-qubit system,could also be realized.

On the other hand, experimental methods for generating an entangledquantum state is also researched. Quantum entanglement is aquantum-mechanical correlation between two systems which can be locallyseparated from each other. More specifically, in a state called a purestate in quantum mechanics, if the overall state of two systems A and Bcannot be expressed in the form of a simple product|Ψ_(AB)>=|ψ_(A)>{circle around (x)}|φ_(B)>, |Ψ_(AB)> is entangled andthe system AB is in a state of quantum entanglement. In this state,neither classical communication between the systems A and B nor a localoperation in each of the systems A and B (unitary transformation of eachof the systems A and B, addition of an auxiliary system, and observationof a local degree of freedom) is possible. Accordingly, it is consideredthat the entangled state has a correlation which cannot be explained byclassical probability theory (see references (12) to (15)).

In a two-qubit system, typical states of quantum entanglement are Bellstates, which are expressed as follows:|Φ^(±)>=(1/√{square root over (2)}) (|00>±|11>)|Ψ^(±)>=(1/√{square root over (2)}) (|01>±|10>)  (1.1)where {|0>, |1>} are orthogonal bases of a two-dimensional Hilbert spacein which two qubits are defined. In Expression (1.1), {|Φ^(±)>, |Ψ^(±)>}are orthogonal bases of a four-dimensional Hilbert space spanned by thetwo qubits, and are therefore called Bell bases. The Bell states play animportant role in quantum teleportation.

One known method for generating two particles in a Bell state isparametric down-conversion, in which a nonlinear optical crystal such asbeta-barium borate (BBO) and LiIO₃ is irradiated with ultraviolet pulsesso that pair creation of two photons whose polarization degrees offreedom are in a Bell state occurs (see reference (16)). In this method,however, the occurrence rate of the down-conversion is determined bytwo-dimensional nonlinear susceptibility χ⁽²⁾, and therefore thegeneration efficiency of Bell photon pairs (Bell pairs) is low.Accordingly, the intensity of the ultraviolet pulses must be increasedin actual experiments. Note that the Bell states can be easily generatedin a system where the controlled-NOT gate transformation can be freelyimplemented. Since it is extremely difficult to realize a controlled-NOTgate, only a method for directly generating the Bell state is describedhere.

The Bell measurement is simultaneous measurement on a 2-qubit systemperformed for distinguishing four Bell bases {|Φ^(±)>, |Ψ^(±)>} from oneanother. In addition to the controlled-NOT gate, the Bell measurement isalso a basic operation in quantum information processing, and isessential in quantum teleportation. Gottesman and Chuang have provedthat the controlled-NOT gate can be implemented by generating aparticular four-qubit state:|χ>=(½)[(|00>+|11>)|00>+(|01>+|10>) |11>]  (1.2)and performing the Bell measurement twice and single-qubit unitarytransformations depending on the result of the Bell measurement (seereference (17)).

In the following description, an interaction-free measurement (IMF) isadopted as the fundamental concept. The IFM is an observation methodformulated by Elitzur and Vaidman and derived to solve the followingproblem. That is, “when there is an object which always absorbs a photonby a strong interaction if the photon comes near enough to the object,how can it be decided whether this object is present or absent withoutcausing it to absorb the photon?” The reason why the photon ispreferably not absorbed by the object is because, for example, there isa risk that the object will explode if it absorbs the photon.

The means by which Elitzur and Vaidman solved this problem will bedescribed below (see also references (18) and (19)). FIG. 20 is adiagram showing an experiment of an interaction-free measurement (IFM)performed by Elitzur and Vaidman. In this experiment, a Mach-Zehnderinterferometer including two beam splitters which act as boundariesbetween an upper path a and a lower path b is used. A state in which asingle photon is present on the path a is expressed as |1>_(a) and astate in which no photon is present on the path a is expressed as and|0>_(a). In addition, an orthogonal relationship _(a)<i|j>_(a)=δ_(ij) issatisfied for any i and j (i,jε{0, 1}). These settings are similar forthe path b. The operations of the two beam splitters B and B′ aredefined as follows:

$\begin{matrix}{B:\left\{ \begin{matrix}\left. {\left. 1 \right\rangle_{a}\left. 0 \right\rangle_{b}}\rightarrow{{\cos\;\theta\left. 1 \right\rangle_{a}\left. 0 \right\rangle_{b}} - {\sin\;\theta\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}}} \right. \\\left. {\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}}\rightarrow{{\sin\;\theta\left. 1 \right\rangle_{a}\left. 0 \right\rangle_{b}} + {\cos\;\theta\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}}} \right.\end{matrix} \right.} & (1.3) \\{B^{\prime}:\left\{ \begin{matrix}\left. {\left. 1 \right\rangle_{a}\left. 0 \right\rangle_{b}}\rightarrow{{\sin\;\theta\left. 1 \right\rangle_{a}\left. 0 \right\rangle_{b}} + {\cos\;\theta\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}}} \right. \\\left. {\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}}\rightarrow{{\sin\;\theta\left. 1 \right\rangle_{a}\left. 0 \right\rangle_{b}} - {\sin\;\theta\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}}} \right.\end{matrix} \right.} & (1.4)\end{matrix}$The upper path a of the interferometer is placed on a point where thepresence/absence of the object is to be determined.

The case in which a photon is injected into the path b from the lowerleft is considered. When nothing is present on the two paths a and b ofthe interferometer, the photon comes out from the path a at the upperright and is detected by a detector D₀. In comparison, when an objectwhich can absorb the photon is present on the upper path a, the objectabsorbs the photon with probability 1 if the photon comes near enough tothe object to cause the interaction. Accordingly, there are threepossibilities:

-   (A) Neither of detectors D₀ and D₁ detects the photon: probability    P_(A)=sin² θ-   (B) Detector D₀ detects the photon: Probability P_(B)=cos^(4 θ)-   (C) Detector D₁ detects the photon: Probability P_(C)=cos² θ sin² θ

(A) means that the photon has been absorbed by the object, and thereforethe condition of IFM is not satisfied. In addition, (B) means that thepresence/absence of the object cannot be determined. (C) means that thepresence of the object is detected without causing the object to absorbthe photon. Elitzur and Vaidman called the operation (C)interaction-free measurement. As used here, the term “interaction-free”describes the case where the photon has not been absorbed by the object.

The efficiency ζ of the IFM is calculated as follows:ζ=P _(C)/(P _(A) +P _(C))  (1.5)The reason why P_(B) is not included in Expression (1.5) is because theexperiment can be retried in the case (B). When θ=π/4, the beamsplitters B and B′ serve as 50—50 beam splitters (beam splitters whosetransmittance and reflectance are both ½), and P_(A), P_(B), P_(C), andζ are determined as P_(A)=½, P_(B)=P_(C)=¼, and ζ=⅓, respectively.Generally, ζ is calculated as follows:ζ=z/(1+z), z=cos² θ, 0≦z≦1  (1.6)FIG. 21 shows a graph of the efficiency ζ versus the reflectance z(0≦z≦1) of the beam splitters, and it is clear from this graph that ζ≦½.

Accordingly, in the method according to Elitzur and Vaidman, theefficiency ζ never exceeds ½. In addition, P_(B) approaches 1 as ζapproaches ½, which means that the number of retries increases. When theobject is present on the path a, the average number of tries taken untilthe measurement finishes by obtaining the result (A) or (C) iscalculated as N=1/(1−P_(B))=1/(1−cos⁴ θ). Accordingly, N diverges toinfinity ( N→∞) when ζ→½ or θ→0. In other words, the number of triesdiverges to infinity as ζ approaches ½.

Kwiat et al. have created a method for causing ζ to asymptoticallyapproach 1 and P_(B) to asymptotically approach 0 (see references (20)and (21)). In the method according to Kwiat et al., an interferometershown in FIG. 22 is used, which includes N beam splitters which act asboundaries between an upper path a and a lower path b. Similar to theabove-described case, a state in which a single photon is present on thepath a is expressed as |1>_(a) and a state in which no photon is presenton the path a is expressed as |0>_(a). In addition, these settings aresimilar for the path b. The operations of the beam splitters B aredefined by Expression (1.3).

A photon is injected through the lower left entrance b. When nothing ispresent on the paths, the wave function of the photon which comes outfrom the k^(th) beam splitter is expressed as follows:sin kθ|1>_(a)|0>_(b)+cos kθ|0>_(a)|1>_(b) , k=0, 1 . . . , N  (1.7)When θ=π/2N, the photon comes out from the upper right exit a of theN^(th) beam splitter with probability 1.

Next, the case is considered in which N identical objects which canabsorb the photon are present on the upper path a at positions behindthe beam splitters. In this case, the photon injected through the lowerleft entrance b cannot pass through the path a since it will be absorbedby the objects if it enters the path a. Accordingly, the probability Pthat the photon will come out from the lower right exit b is calculatedas the product of the reflectances of the beam splitters (P=cos^(2N) θ)When N increases to infinity, P approaches 1:

$\begin{matrix}{{\lim\limits_{N\rightarrow\infty}\; P} = {{\lim\limits_{N\rightarrow\infty}\;{\cos^{2N}\left( \frac{\pi}{2N} \right)}} = {{\lim\limits_{N\rightarrow\infty}\left\lbrack {1 - \frac{\pi^{2}}{4N} + {O\left( \frac{1}{N^{2}} \right)}} \right\rbrack} = 1}}} & (1.8)\end{matrix}$Accordingly, the efficiency ζ (=P) in detecting the objects by the IFMapproaches 1 when N→∞.

As is clear from the above-discussion, the interferometer according toKwiat et al. changes the direction in which the photon injected from thelower left travels as follows, at least with probability P:

-   (1) If no absorbing object is present in the interferometer, the    photon comes out from the upper right exit a.-   (2) If the absorbing objects are present in the interferometer, the    photon will come out from the lower right exit b.    In addition, P approaches 1 as N increases. In the following    description, the interferometer shown in FIG. 22 proposed by Kwiat    et al. is called an IFM interferometer.

The documents listed below are incorporated herein by reference:

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In the field of quantum information processing such as quantumcomputing, quantum communication, and quantum cryptography, the threemost important basic operations are: generation of the Bell state, theBell measurement, and the controlled-NOT gate transformation. Thesethree operations are not independent from one another but are closelyrelated to one another, and there are demands for these operations.

As described above, one known method for generating two particles in aBell state is parametric down-conversion, in which a nonlinear opticalcrystal is irradiated with ultraviolet pulses so that pair creation oftwo photons whose polarization degrees of freedom are in a Bell stateoccurs. In this method, however, the occurrence rate of thedown-conversion is determined by the two-dimensional nonlinearsusceptibility χ⁽²⁾, and therefore the generation efficiency of Bellphoton pairs is low. Accordingly, the intensity of the ultravioletpulses must be increased in actual experiments.

The Bell measurement is an essential technique in quantum teleportation,and it is an important objective in quantum information processing tocreate a simple method for the Bell measurement.

The controlled-NOT gate is regarded as an essential technique to realizea quantum computer. More specifically, a U(2) transformation gate for asingle qubit and a controlled-NOT gate form a universal set of gates forquantum computation, and it is known that any kind of operation onqubits can be implemented by combining these gates. Accordingly, torealize a controlled-NOT gate is one of the most important objectives inquantum information processing. If a controlled-NOT gate were realized,generation of the Bell state and the Bell measurement could also berealized.

However, it is difficult to realize a controlled-NOT gate which producesa quantum correlation between two qubits. Although a method using cavityQED and other methods have been proposed, as described above, thesemethods require highly advanced experimental techniques, and are notexpected to be put into practical use in the near future.

On the other hand, Gottesman and Chuang have proved that acontrolled-NOT gate can be implemented by generating a particularfour-qubit state |χ> and performing the Bell measurement twice andsingle-qubit unitary transformations depending on the result of the Bellmeasurement. This means that although it is difficult to implement thecontrolled-NOT gate directly, it can be implemented indirectly if theBell measurement can be performed easily.

Thus, the Bell measurement and the controlled-NOT gate are closelyrelated to each other, and there is a requirement to create a simplemethod for the Bell measurement and to thereby realize a controlled-NOTgate.

SUMMARY OF THE INVENTION

Accordingly, an object of the present invention is to easily generate aBell state using an interaction-free measurement (IFM), to perform theBell measurement, which is simultaneous measurement on a two-particlesystem, and to implement a controlled-NOT gate, which is unitarytransformation of a two-particle system.

In addition, another object of the present invention is to provide anapproximate evaluation method for the fidelity of a logic gate using theinteraction-free measurement (IFM).

According to one aspect of the present invention, an apparatus forgenerating a quantum state of a two-qubit system including two qubits,each qubit being represented by a particle which invariably travelsthrough one of two paths, includes an input unit for receiving twoparticles having no correlation with each other; and a quantum gatecomposed of an interferometer for implementing an interaction-freemeasurement, the quantum gate generating a Bell state with asymptoticprobability 1.

According to another aspect of the present invention, a Bell measurementapparatus for a two-qubit system including two qubits, each qubit beingrepresented by a particle which invariably travels through one of twopaths, includes an input unit for receiving the state of the two-qubitsystem; at least one quantum gate composed of an interferometer forimplementing an interaction-free measurement; an observation unit forobserving the quantum gate after the state of the two-qubit system hasbeen processed by the quantum gate; and an identifying unit whichperforms a Bell measurement for selecting the state of the two-qubitsystem from among the Bell bases on the basis of the result of theobservation.

According to still another aspect of the present invention, a method forevaluating the fidelity of a quantum gate composed of an interferometerfor implementing an interaction-free measurement, includes the steps ofdetermining an absorption probability with which a first particleabsorbs a second particle in the interferometer; calculating, if theabsorption probability is less than 1, an approximate fidelity of thequantum gate under the condition that the number of times the secondparticle hits beam splitters in the interferometer is sufficientlylarge.

Other objectives and advantages besides those discussed above shall beapparent to those skilled in the art from the description of a preferredembodiment of the invention which follows. In the description, referenceis made to accompanying drawings, which form a part thereof, and whichillustrate an example of the invention. Such example, however, is notexhaustive of the various embodiments of the invention, and thereforereference is made to the claims which follow the description fordetermining the scope of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram showing a quantum circuit for generating a Bellstate.

FIG. 2 is a diagram showing an IFM gate proposed by Kwiat et al.

FIG. 3 is table showing a state transformation performed by the IFMgate.

FIG. 4 is a diagram showing an interferometer according to Kwiat et al.,which uses a positron-electron pair.

FIG. 5 is a diagram showing a quantum circuit for generating andoutputting a GHZ state.

FIG. 6 is an energy level diagram of an auxiliary atom used forgenerating of a two-photon Bell state.

FIG. 7 is a diagram showing a quantum circuit for generating andoutputting the Bell state from two photons having no correlation.

FIG. 8 is a diagram showing a quantum circuit for performing a Bellmeasurement using the IFM gate.

FIG. 9 is a table showing a state transformation performed by anexpanded IFM gate which allows dissipation in a system.

FIG. 10 is a table showing a state transformation performed when atwo-qubit state is input to the quantum circuit shown in FIG. 8.

FIG. 11 is a table showing a random permutation of Bell-basis vectors.

FIG. 12 is a diagram showing a quantum circuit for performing the Bellmeasurement using IFM gates.

FIG. 13 is a table showing the results of a state transformationobtained at t=T₁ and t=T₂ in FIG. 12.

FIG. 14 is a diagram showing a quantum circuit which provides aparticular four-qubit entangled state.

FIG. 15 is a diagram showing the construction of a controlled-NOT gateaccording to Gottesman and Chuang.

FIG. 16 is a diagram showing the exchange of wave functions between apositron and an electron.

FIG. 17 is a diagram showing an apparatus for generating the Bell stateof a positron-electron pair.

FIG. 18 is a graph of the fidelity of the IFM gate plotted as a functionof the number N of beam splitters and the probability η that anabsorbing object will fail to absorb a photon.

FIG. 19 is a graph of the number N of beam splitters required forobtaining an IFM gate with a desired fidelity plotted as a function ofthe probability η that the absorbing object will fail to absorb thephoton.

FIG. 20 is a diagram showing the experiment of an interaction-freemeasurement.

FIG. 21 is a graph of the efficiency ζ of the interaction-freemeasurement plotted as a function of the reflectance z of beamsplitters.

FIG. 22 is a diagram showing an interferometer according to Kwiat et al.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Embodiments of the present invention will be described below withreference to the accompanying drawings.

A Bell-state-generating apparatus, a Bell measurement apparatus, and acontrolled-NOT gate transformation apparatus include a quantum gatecomposed of an interferometer according to Kwiat et al., and the quantumgate operates with probability 1 when N→∞, where N is the number of beamsplitters included in the interferometer according to Kwiat et al.

Apparatuses described below implement a logic gate using aninteraction-free measurement to realize the generation of a Bell state,a Bell measurement, which is simultaneous measurement on a two-particlesystem, and a controlled-NOT gate, which is unitary transformation of atwo-particle system.

In addition, according to a method for evaluating the fidelity of thequantum gate described below, an approximate evaluation of the fidelityof the quantum gate composed of the interferometer according to Kwiat etal. is performed under the condition that a particle absorptionprobability is fixed and the number of times the second particle hitsbeam splitters in the interferometer is sufficiently large.

First Embodiment

In a first embodiment, a Bell-state-generating apparatus will bedescribed.

In the above-described interferometer according to Kwiat et al., themoving direction of the particle B (photon) is changed depending on thepresence/absence of the particle A (absorbing object). This can beinterpreted as writing information of the particle A to the particle B.Accordingly, in the present embodiment, the interferometer according toKwiat et al. is regarded as a quantum gate between the two particles. Inaddition, in the interferometer according to Kwiat et al., the absorbingobject is regarded as a classical particle which is either present orabsent in the interferometer. In comparison, in the present embodiment,the absorbing object is regarded as a quantum particle which can be in asuperposition of present and absent states in the interferometer.

In addition, in the present embodiment, the particle A is input to thequantum gate while it is in a quantum superposition of the present andabsent states in a chamber of a cavity, so that the Bell state of thetwo particles A and B is output.

In the following description, a Bell-state-generating apparatus using anelectron-positron pair will be explained.

The interaction-free measurement (IFM) is an experiment using aninterferometer which includes a cavity and beam splitters sectioning thecavity into two chambers. Two particles consisting of a particle B and aparticle A which absorbs the particle B if the particles A and B comenear enough to each other are input to different chambers, and theparticle B is caused to successively hit the beam splitters, so that thetransmitted wave component of the wave function of the particle Btravels back and forth between the two chambers. The particletransmittance of the beam splitters is set low so that the probabilityamplitude of the state in which the particle B is absorbed by theparticle A by entering the same chamber with the particle A when theparticle B particle hits one of the beam splitters is set small. As thenumber of times the particle B hits the beam splitters increases and thetransmittance of the beam splitters reduces, the probability that theparticle A will absorb the particle B approaches zero. Accordingly, theparticle B is put into different chambers depending on whether or notthe particle A is input to the cavity. As described above, in the IFMinterferometer, one of two particles travels through different pathsdepending on the presence/absence of the other particle, and thus theIFM interferometer functions as a quantum gate.

In the interferometer according to Kwiat et al. shown in FIG. 22, thephoton comes out from the interferometer through different paths a and bdepending on the presence/absence of the absorbing objects withasymptotic probability 1 when N→∞ (N is the number of beam splitters).This can be interpreted as writing information of the absorbing objectsto the photon. In addition, in the IFM according to Kwiat et al.,dissipation (annihilation) of the photon does not occur when N→∞, andtherefore no state contraction occurs. Accordingly, the quantum statewill not be destroyed in the process of IFM, which means that the objectcan be regarded not only as a classical object but also as a quantumobject. More specifically, in the above-described known technique, eachof the absorbing objects is regarded as a classical object which iseither present or absent in the interferometer. However, in the presentembodiment, an absorbing object can be in a superposition of twoorthogonal states, that is, the present and absent states, isconsidered, and this object is input to the interferometer. This meansthat the absorbing object is regarded as a quantum object. Thesubstitution of a classical absorbing object with a quantum object inthe IFM according to Elitzur and Vaidman has been discussed by Hardy (L.Hardy, “Quantum mechanics, local realistic theories, andLorentz-invariant realistic theories”, Phys. Rev. Lett. 68, 2981–2984(1992).)

From the above discussion, it is expected that the IFM interferometeraccording to Kwiat et al. will function as a quantum gate. Accordingly,the interferometer shown in FIG. 22 is represented by the symbol shownin FIG. 2. The entrances a and b at the upper left and the lower left inFIG. 22 correspond to entrances a and b, respectively, in FIG. 2, andthe exits a and b at the upper right and the lower right in FIG. 22correspond to exits a′ and b′, respectively, in FIG. 2. In FIG. 2, anabsorbing object is input from x and is output from x′. As describedabove in the section of the related art, no photon is input from theentrance a in FIG. 2 in the IFM, and therefore the path extending fromthe entrance a is shown by a dashed line and a, filled rectangle isshown adjacent to the entrance a in the symbol representing the gate.The symbol shown in FIG. 2 is hereafter called an IFM gate. In addition,a section including the paths x and x′ and a section including the pathsa, b, a′, and b′ are sometimes called a control section and a targetsection, respectively.

A state transformation obtained by the IFM gate shown in FIG. 2 when N→∞is shown in FIG. 3. In the table of FIG. 3, the first line correspondsto the case in which no absorbing object is provided and a photon isinput to the path b, and the second line corresponds to the case inwhich an absorbing object is input to the path x and a photon is inputto the path b. In the following description, it is assumed that N isincreased to infinity (N→∞) in the IFM gate and the IFM gate performsthe transformation shown in FIG. 3. The IFM gate also performs thelinear transformation shown in FIG. 3 when an object in a superpositionof |0>_(x) and |1>_(x) is input to the path x.

For convenience, an electron and a positron are considered in place ofthe photon and the absorbing object, respectively, in the followingdiscussion (the electron and the positron are hereafter sometimesexpressed as e⁻ and e⁺, respectively). When the electron and thepositron come near enough to each other, a photon is generated by pairannihilation. Here, it is assumed that this reaction occurs withprobability 1. This reaction can be interpreted as the absorption of theelectron by the positron. In addition, when a suitable potential barrieris used, beam splitters and mirrors for the electron and the positroncan be obtained. Accordingly, it is possible to form an interferometersimilar to that shown in FIG. 22 for the electron and the positron.

It is to be noted that, in order to form an IFM gate for the electronand the positron, the speeds of the electron and the positron and thepaths along which they travel must be adjusted such that the twoparticles approach each other at positions corresponding to thepositions where the absorbing objects are placed in the cavity in FIG.22. The electron and the positron considered here are quantum objects,and each of them must be regarded as a wave packet having fluctuations(expansions) Δ{right arrow over (x)} and Δ{right arrow over (p)}(|Δ{right arrow over (x)}||Δ{right arrow over (p)}|˜

/2 is satisfied by the uncertainty principle). In order for the pairannihilation of the electron and the positron to occur, the distancebetween the two particles must be Δr or less at time t (Δr is aparticular range of Coulomb interaction). In the reaction consideredhere, |Δ{right arrow over (x)}|<<Δr is assumed. Accordingly, it is notnecessary to take into account the quantum mechanical expansion of thewave packet when the two particles approach each other, and thereforethe electron and the positron can be regarded as point particles.

The actual construction of the interferometer is shown in FIG. 4. FIG. 4shows an interferometer according to Kwiat et al. to which a positronand an electron generated by an accelerator are input. Theinterferometer includes a vacuum vessel containing mirrors and beamsplitters composed of metal plates providing a suitable potentialbarrier. In FIG. 4, paths x, a, and b at the left and paths x′, a′, andb′ at the right respectively correspond to the paths x, a, b, x′, a′,and b′ shown in FIG. 2. The interferometer shown in FIG. 4 includes fivebeam splitters, and the circles show the positions where thepositron-electron pair annihilation occurs. The speeds of the positronand the electron must be adjusted such that they approach each other atthe positions shown by the circles.

FIG. 1 shows an apparatus which generates and outputs a Bell state whena positron and an electron having no correlation with each other areinput. The apparatus shown in FIG. 1 is sometimes called a quantumcircuit since it is a combination of gates which operatequantum-theoretically. The operation of a beam splitter H shown in FIG.1 is defined as follows:

$\begin{matrix}{H:\left\{ \begin{matrix}\left. {\left. 0 \right\rangle_{x}\left. 1 \right\rangle_{y}}\rightarrow{\left( {1/\sqrt{2}} \right)\left( {{\left. 0 \right\rangle_{x}\left. 1 \right\rangle_{y}} + {\left. 1 \right\rangle_{x}\left. 0 \right\rangle_{y}}} \right)} \right. \\\left. {\left. 1 \right\rangle_{x}\left. 0 \right\rangle_{y}}\rightarrow{\left( {1/\sqrt{2}} \right)\left( {{\left. 0 \right\rangle_{x}\left. 1 \right\rangle_{y}} - {\left. 1 \right\rangle_{x}\left. 0 \right\rangle_{y}}} \right)} \right.\end{matrix} \right.} & (2.1)\end{matrix}$This transformation is called a Hadamard transformation. When a positronis input to the beam splitter H from the path y, it is output from thetwo paths in a superposition with an amplitude of 1/√{square root over(2)}. The IFM gate operates as described above with reference to FIGS.22, 2, and 3.

The operation of the quantum circuit shown in FIG. 1 will be describedbelow. Initially, a positron is input to the path y and an electron isinput to the path b. The state in which a single photon is present inthe path x is expressed as |1>_(x) and state in which no photon ispresent in the path x is expressed as |0>_(x). In addition, it isassumed that an orthogonal relationship _(x)<i|j>_(x)=δ_(ij) issatisfied for any i and j (i,jε{0, 1}). These settings are similar forthe paths y, a, and b. In FIG. 1, the state changes from the left to theright as follows:

$\begin{matrix}{\left. 0 \right\rangle_{x}\left. 1 \right\rangle_{y}\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}} & (2.2) \\\left. {H\text{:}}\rightarrow{\left( {1/\sqrt{2}} \right)\left( {{\left. 0 \right\rangle_{x}\left. 1 \right\rangle_{y}} + {\left. 1 \right\rangle_{x}\left. 0 \right\rangle_{y}}} \right)\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}} \right. & \; \\\left. {{IFM}\mspace{14mu}{gate}\text{:}}\rightarrow{\left( {1/\sqrt{2}} \right)\left( {{\left. 0 \right\rangle_{x}\left. 1 \right\rangle_{y}\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}} + {\left. 1 \right\rangle_{x}\left. 0 \right\rangle_{y}\left. 1 \right\rangle_{a}\left. 0 \right\rangle_{b}}} \right)} \right. & \;\end{matrix}$Logical ket vectors of the positron and the electron are defined asfollows:| 0>₊=|0>_(x)|1>_(y), | 1>₊=|1>_(x)|0>_(y)  (2.3)| 0>⁻=|0>_(a)|1>_(b), | 1>⁻=|1>_(a)|0>_(b)  (2.4)where an orthogonal relationship _(α)<ī| j>₆₂ =Δ_(αβ)Δ_(ij) is satisfiedfor any α and β (α, βε{+,−}) and for any i and j (i, jε{0,1}).Accordingly, the transformation of Expression (2.2) can be rewritten asfollows:

$\begin{matrix}{\left. {\left. \overset{\_}{0} \right\rangle_{+}\left. \overset{\_}{0} \right\rangle_{-}}\rightarrow\left. \Phi^{+} \right\rangle \right. = {\left( {1/\sqrt{2}} \right)\left( {{\left. \overset{\_}{0} \right\rangle_{+}\left. \overset{\_}{0} \right\rangle_{-}} + {\left. \overset{\_}{1} \right\rangle_{+}\left. \overset{\_}{1} \right\rangle_{-}}} \right)}} & (2.5)\end{matrix}$This means that the Bell state is generated from two particles having nocorrelation.

The method for obtaining the logical ket vectors {| 0 >, | 1 >} of aqubit using two paths as in Expressions (2.3) and (2.4) is called adual-rail representation (I. L. Chuang and Y. Yamamoto, “Simple quantumcomputer”, Phys. Rev. A 52, 3489–3496 (1995).) In thisqubit-representing method, it is ensured that a particle is alwayspresent in one of the two paths. In addition, by placing two beamsplitters B and B′ between the two paths as shown in FIG. 20, a desiredunitary transformation can be implemented on a two-dimensional Hilbertspace spanned by {| 0>, | 1>}. Although |Φ⁺> is generated in Expressions(2.2) and (2.5), |Φ⁻> and |Ψ^(±)> can also be obtained by adding a beamsplitter between two paths forming a qubit.

In the quantum circuit shown in FIG. 1, the number of electrons andpositrons is maintained between the initial and final states, and nopair annihilation occurs. In this view, the generation of the Bell stateaccording to Expressions (2.2) and (2.5) can be regarded as aninteraction-free process. In addition, since the probability P that theIFM gate will operate correctly approaches 1 when N→∞, the fidelityF=|<Φ⁺φ>|² of the state |φ> generated by Expressions (2.2) and (2.5)also approaches 1 when N→∞.

A GHZ state (1/√{square root over (2)})(|000>+|111>) can also begenerated by a similar method. FIG. 5 shows a quantum circuit whichgenerates and outputs a GHZ state when two positrons and an electronhaving no correlation with one another are input. The quantum circuitperforms the following transformation:

$\begin{matrix}\left. {\left. \overset{\_}{0} \right\rangle_{+}\left. \overset{\_}{0} \right\rangle_{-}\left. \overset{\_}{0} \right\rangle_{+}}\rightarrow{\left( {1/\sqrt{2}} \right)\left( {{\left. \overset{\_}{0} \right\rangle_{+}\left. \overset{\_}{0} \right\rangle_{-}\left. \overset{\_}{0} \right\rangle_{+}} + {\left. \overset{\_}{1} \right\rangle_{+}\left. \overset{\_}{1} \right\rangle_{-}\left. \overset{\_}{1} \right\rangle_{+}}} \right)} \right. & (2.6)\end{matrix}$

In the present embodiment, the case in which a positron and an electronare used as two particles is described. However, any kinds of particlesmay be used as long as one of them absorbs the other when they approacheach other.

Second Embodiment

In a second embodiment, an apparatus for generating the Bell state usinga pair of photons will be described.

Generation of the Bell state using photons instead of theelectron-positron pair, which causes pair annihilation, as qubits willbe described below. In this case, an atom is required as an object whichabsorbs the photons. In the present embodiment, an experiment isconsidered in which only a Rabi oscillation and beam splitters for thephotons are used as elements. The reason that these two elements areused is because they are frequently used in cavity-QED experiments. FIG.6 is an energy level diagram of an auxiliary atom used for generating atwo-photon Bell state by the IFM. The atom has three levels: a groundstate g₀, a first excited state e₁, and a second excited state e₂, andan energy difference between e₁ and g₀ and that between e₂ and g₀ are

ω₁ and

ω₂, respectively. It is assumed that ω₂>ω₁ is satisfied,

ω₁,

ω₂, and

(ω₂−ω₁) are sufficiently large, and τ₁>>τ₂ is satisfied, where τ₁ is thespontaneous emission lifetime of e₁→g₀+

ω₁ and τ₂ is the time required to excite the atom from g₀ to e₂ bycausing it to absorb a photon with an angular frequency ω₂.

When an electric field with an angular frequency slightly shifted fromω₁ (laser pulse ω=ω₁−Δω, 0<|Δω|<<ω₁) is applied to the atom, Rabioscillation between the ground state g₀ and the first excited state e₁occurs (R. Loudon, “The Quantum Theory of Light”, second edition,(Oxford, Oxford University Press, 1983)). When a photon with an angularfrequency ω₂ is regarded as a qubit, the atom can absorb the photon ifthe atom is in the state g₀ but cannot absorb the photon if the atom isin the state e₁, and this can be used for performing the IFM. Here, itis assumed that the photon with the angular frequency ω₂ is absorbed bythe atom with probability 1 if the photon is incident on the atom whilethe atom is in the state g₀.

FIG. 7 shows a quantum circuit which uses a three-level atom as anauxiliary system to generate and output the Bell state from two photonshaving no correlation. The atom is input to a path x while it is in thestate |e₁>, and photons with the angular frequency ω₂ are input to pathsb and d. The state of the atom in the path x is expressed as follows:|e₁>=|0>_(x), |g₀>=|1>_(x)  (3.1)The atom traveling through the path x is irradiated with a combinationof suitable laser pulses so that the Rabi oscillation occurs, and theHadamard transformation H is thereby implemented as follows:

$\begin{matrix}{\left. {H\text{:}\mspace{14mu}\left. 0 \right\rangle_{x}}\rightarrow{\left( {1/\sqrt{2}} \right)\left( {\left. 0 \right\rangle_{x} + \left. 1 \right\rangle_{x}} \right)} \right.,\left. \left. 1 \right\rangle_{x}\rightarrow{\left( {1/\sqrt{2}} \right)\left( {\left. 0 \right\rangle_{x} - \left. 1 \right\rangle_{x}} \right)} \right.} & (3.2)\end{matrix}$The IFM gates perform the transformation defined by the table of FIG. 3.

In FIG. 7, the state of the overall system changes from the left to theright as follows:

$\begin{matrix}{\left. 0 \right\rangle_{x}\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}\left. 0 \right\rangle_{c}\left. 1 \right\rangle_{d}} \\\left. {H\text{:}}\mspace{14mu}\rightarrow{\left( {1/\sqrt{2}} \right)\left( {\left. 0 \right\rangle_{x} + \left. 1 \right\rangle_{x}} \right)\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}\left. 0 \right\rangle_{c}\left. 1 \right\rangle_{d}} \right.\end{matrix}$

-   First IFM gate:

$\left. \rightarrow{\left( {1/\sqrt{2}} \right)\left( \left. {{\left. 0 \right\rangle_{x}\left. 1 \right\rangle_{a}\left. 0 \right\rangle_{b}} + {\left. 1 \right\rangle_{x}\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{x}\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}}} \right| \right)\mspace{14mu}\left. 0 \right\rangle_{c}\left. 1 \right\rangle_{d}} \right.$

-   Second IFM gate:

$\begin{matrix}\left. \rightarrow{\left( {1/\sqrt{2}} \right)\left( {{\left. 0 \right\rangle_{x}\left. 1 \right\rangle_{a}\left. 0 \right\rangle_{b}\left. 1 \right\rangle_{c}\left. 0 \right\rangle_{d}} + {\left. 1 \right\rangle_{x}\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}\left. 0 \right\rangle_{c}\left. 1 \right\rangle_{d}}} \right)} \right. & (3.3) \\{= {\left( {1/\sqrt{2}} \right)\left( {{\left. 0 \right\rangle_{x}\left. \overset{\_}{1} \right\rangle_{\omega}\left. \overset{\_}{1} \right\rangle_{\omega}} + {\left. 1 \right\rangle_{x}\left. \overset{\_}{0} \right\rangle_{\omega}\left. \overset{\_}{0} \right\rangle_{\omega}}} \right)}} & \; \\{\left. {H\text{:}}\mspace{14mu}\rightarrow{\left( {1/2} \right)\left\lbrack {{\left( {\left. 0 \right\rangle_{x} + \left. 1 \right\rangle_{x}} \right)\left. \overset{\_}{1} \right\rangle_{\omega}\left. \overset{\_}{1} \right\rangle_{\omega}} + {\left( {\left. 0 \right\rangle_{x} - \left. 1 \right\rangle_{x}} \right)\left. \overset{\_}{0} \right\rangle_{\omega}\left. \overset{\_}{0} \right\rangle_{\omega}}} \right\rbrack} \right\rbrack\mspace{14mu}} & \; \\{= {\left( {1/2} \right)\left\lbrack {{\left. 0 \right\rangle_{x}\left( {{\left. \overset{\_}{0} \right\rangle_{\omega}\left. \overset{\_}{0} \right\rangle_{\omega}} + {\left. \overset{\_}{1} \right\rangle_{\omega}\left. \overset{\_}{1} \right\rangle_{\omega}}} \right)} - {\left. 1 \right\rangle_{x}\left( {{\left. \overset{\_}{0} \right\rangle_{\omega}\left. \overset{\_}{0} \right\rangle_{\omega}} - {\left. \overset{\_}{1} \right\rangle_{\omega}\left. \overset{\_}{1} \right\rangle_{\omega}}} \right)}} \right\rbrack}} & \;\end{matrix}$Then, the atom is observed in the basis {|0>_(x), |1>_(x)}={|e₁>, |g₀>}.If |0>_(x)=|e₁> is observed, the two photons are projected into|Φ⁺>=(1/√{square root over (2)}) (| 0>_(ω)| 0>_(ω)+| 1>_(ω)| 1>_(ω)). If|1>_(x)=|g₀> is observed, the two photons are projected into|Φ⁻>=(1/√{square root over (2)}) (| 0>_(ω)| 0>_(ω)−| 1>_(ω)| 1>_(ω)).Accordingly, the Bell state of two photons is generated.

Some methods for producing quantum correlation (entanglement) betweentwo photons using the technique of cavity-QED have been proposed (Q. A.Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble,“Measurement of conditional phase shifts for quantum logic”, Phys. Rev.Lett. 75, 4710–4713 (1995); C. Monroe, D. M. Meekhof, B. E. King, W. M.Itano, and D. J. Weinland, “Demonstration of a fundamental quantum logicgate”, Phys. Rev. Lett. 75, 4714–4717 (1995)). However, the principlesof these methods are different from that of the method according to thepresent embodiment.

Third Embodiment

In a third embodiment, a first example of an apparatus for performingthe Bell measurement will be described.

In a Bell measurement apparatus according to the present embodiment, astate to be observed is input to a quantum circuit obtained by combiningthe above-described quantum gate and beam splitters, and an adequateobservation is performed after the state has been processed by thecircuit. Then, the Bell basis of the observed state is selected fromamong the four Bell bases depending on the result of the observation.

A method for distinguishing two-qubit states {|Φ^(±)>, |Ψ^(±)>} of asystem consisting of a positron and an electron from one another usingthe IFM gate will be described below (the Bell states obtained by athree-level atom and photons can also be distinguished from one anotherby a similar method). FIG. 8 shows a quantum circuit for implementing aBell measurement using the IFM gate. One of the states {|Φ^(±)>,|Ψ^(±)>} is input to this quantum circuit. Although no particle is inputto the path a of the IFM gate in the above-described embodiments, thecase in which an electron is input to the path a is also considered inthe present embodiment (the input is expanded compared to FIG. 3). Thethus expanded IFM gate which allows dissipation in the system performs atransformation shown in FIG. 9 (it is assumed that the number N of beamsplitters included in the IFM interferometer is increased to infinity(N→∞)). Note that the wave function is multiplied by a phase factor (−1)if an electron is input to the path a while no positron is input to thepath x. In addition, if a positron is input to the path x and anelectron is input to the path a, pair annihilation of the electron andthe positron occurs and a photon γ is generated (e⁺e⁻→γ). The symbol|γ>_(xa) in the fifth row of the table of FIG. 9 represents thisprocess. When an atom and photons are input instead of the electron andthe positron, as in the second embodiment, |γ>_(xa) in the fifth rowmeans that the atom is in the second excited state e₂. In either case,it means that dissipation has occurred in the system and the systemcannot function as a quantum gate. In this view, the IFM gate is notunitary. In the following description, it is assumed that N is increasedto infinity (N→∞) in the IFM gate and the IFM gate performs thetransformation shown in FIG. 9.

When the two-qubit states {| 0>₊| 0>⁻, | 0>₊| 1>⁻, | 1>₊| 0>⁻, | 1>₊|1>⁻} are input to the quantum circuit shown in FIG. 8, a statetransformation shown in FIG. 10 is performed. Note that | 022₊=|0>_(x)|1>_(y), | 1>₊=|1>_(x)|0>_(y), | 0>⁻=|0>_(a)|1>_(b), and | 1>⁻=|1>_(a)|0>_(b) are defined as in Expressions (2.3) and (2.4), and thestate transformation shown in FIG. 10 is arranged in the order of | 0>₊|0>⁻, | 1>₊| 1>⁻, | 0>₊| 1>⁻, and | 1>₊| 0>⁻ from the top. Since one ofthe states {|Φ^(±)>, |Ψ^(±)>} is input, when the path b′ is observed inthe basis {|0>_(b′), |1>_(b′)}, |0>_(b′) is obtained for |Φ^(±)> and|1>_(b′) is obtained for |Ψ^(±)>. Thus, |Φ^(±)> and |Ψ^(±)> can bedistinguished from each other by observing the path b′.

The case is considered in which one of |Ψ⁺> and |Ψ⁻> is input. When theparticles are input to the quantum circuit shown in FIG. 8 and |1>_(b′)is obtained as a result of observation of the path b′, the state changesas follows:

$\begin{matrix}{❘{\Psi^{\pm}>={\left( {1/\sqrt{2}} \right)\left( {{\left. \overset{\_}{0} \right\rangle_{+}\left. \overset{\_}{1} \right\rangle_{-}} \pm {\left. \overset{\_}{1} \right\rangle_{+}\left. \overset{\_}{0} \right\rangle_{-}}} \right)}}} \\{= {\left( {1/\sqrt{2}} \right)\left( {{\left. 0 \right\rangle_{x}\left. 1 \right\rangle_{y}\left. 1 \right\rangle_{a}\left. 0 \right\rangle_{b}} \pm {\left. 1 \right\rangle_{x}\left. 0 \right\rangle_{y}\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}}} \right)}}\end{matrix}$IFM gate: →(1/√{square root over (2)})(−|0>_(x′)|1>_(y′)±|1>_(x′)|0>_(y′)|)0>_(a′)|1>_(b′)

-   Observation at Path b′:

$\begin{matrix}{{->{\left( {1/\sqrt{2}} \right)\left( {{{- \left. 0 \right\rangle_{x^{\prime}}}\left. 1 \right\rangle_{y^{\prime}}} \pm {\left. 1 \right\rangle_{x^{\prime}}\left. 0 \right\rangle_{y^{\prime}}}} \right)\left. 0 \right\rangle_{a^{\prime}}}} = {\left( {1/\sqrt{2}} \right)\left( {{- \left. \overset{\_}{0} \right\rangle_{+}} \pm \left. \overset{\_}{1} \right\rangle_{+}} \right)\left. 0 \right\rangle_{a^{\prime}}}} & (4.1)\end{matrix}$Then, the positron in the paths x and y is input to the beam splitter Hdefined by Expression (2.1). The operation of the beam splitter H can berewritten using the basis {| 0>, | 1>} as follows:H: | 0>→(1/√{square root over (2)})(| 0>+| 1>), | 1>→(1/√{square rootover (2)})(| 0>−| 1>)  (4.2)Accordingly, |Ψ⁺> or |Ψ⁻> which is input to the quantum circuit iseventually converted as follows:|Ψ⁺>→−| 1>₊|0>_(a′), |Ψ⁻>→−| 0>₊|0>_(a′)  (4.3)Thus, |Ψ⁺> and |Ψ⁻> can be distinguished from each other by observingthe paths x and y in the basis {| 0>₊, | 1>₊}.

When one of |Φ⁺> and |Φ⁻> is input to the quantum circuit, dissipationoccurs in the system due to the pair annihilation of thepositron-electron pair, and further quantum operation is impossible.Therefore, if |0>_(b′) is observed at the path b′, |Φ⁺> or |Φ⁻> israndomly determined by a classical coin-toss method or the like.

As a result, |Ψ⁺> and |Ψ⁻> can be distinguished from each other withprobability 1, and |Φ⁺> and |Φ⁻> can be distinguished from each otherwith probability ½. In quantum teleportation, it is necessary todistinguish the four basis vectors {|Φ^(±)>, |Ψ^(±)>} in the followingstate:

$\begin{matrix}\left. {{\left. \psi \right\rangle \otimes \left. \Phi^{+} \right\rangle} = {{{\left( {1/2} \right)\left\lbrack {\left. \Phi^{+} \right\rangle \otimes} \right.}\psi} > {{{{+ \left. \Phi^{-} \right\rangle} \otimes \sigma_{z}}\left. \psi \right\rangle} + {{\left. \Psi^{+} \right\rangle \otimes \sigma_{x}}\left. \psi \right\rangle} + {i{\left. \Psi^{-} \right\rangle \otimes \sigma_{y}}\left. \psi \right\rangle}}}} \right\rbrack & (4.4)\end{matrix}$where |ψ> is an arbitrary single-qubit state. In the expression shownabove, the four Bell-basis vectors are superposed with the sameprobability amplitude. When the above-described Bell measurement usingthe IFM gate is applied, quantum teleportation can be implemented withmaximum probability ¾.

Next, a method for observing an arbitrary two-qubit state:|Σ>=c ₀₀|Φ⁺ >+c ₀₁|Φ⁻ >+c ₁₀|Ψ⁺ >+c ₁₁|Ψ⁻>  (4.5)where c_(ij) εC(complex number), ∀i, jε{0,1}, and

${{\sum\limits_{i,{j \in {\{{0,1}\}}}}^{\;}\;{c_{ij}}^{2}} = 1},$in the Bell bases {|Φ^(±)>, |Ψ^(±)>} is considered. In theabove-described method, |Ψ⁺> and |Ψ⁻> can be distinguished from eachother with probability 1 and |Φ⁺> and |Φ⁻> can be distinguished fromeach other with probability ½. Accordingly, the Bell bases can bedistinguished from one another with average maximum probability ¾ byrandomly permuting the basis vectors {|Φ^(±)>, |Ψ^(±)>} and performingthe observation.

An example of permutation of the Bell bases will be described below.First, an SU(2) rotation operator around the x, y, and z axes is definedas follows:R _(k) (θ)=exp[−i(θ/2)σ_(k) ], kε{x, y, z}, 0≦θ<4π  (4.6)where σ_(k)(kε{x, y, z}) represent Pauli matrices:

$\begin{matrix}{{\sigma_{x} = \begin{pmatrix}0 & 1 \\1 & 0\end{pmatrix}},{\sigma_{y} = \begin{pmatrix}0 & {- i} \\i & 0\end{pmatrix}},{{{and}\mspace{14mu}\sigma_{z}} = \begin{pmatrix}1 & 0 \\0 & {- 1}\end{pmatrix}}} & (4.7)\end{matrix}$In addition, the following equations are satisfied:

$\begin{matrix}{{{R_{k}(\pi)} = {{- i}\;\sigma_{k}}},{{R_{k}\left( {\pi/2} \right)} = {\left( {1/\sqrt{2}} \right)\left( {{{- i}\;\sigma_{k}} + I} \right)}},{I = \begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix}},{k \in \left\{ {x,y,z} \right\}}} & (4.8)\end{matrix}$Accordingly, from calculations such as:[R_(x)(π/2){circle around (x)}R_(x)(π/2)][R_(y)(π){circle around(x)}I]|Ψ⁺>=−|Φ⁻>,  (4.9)the following relationship can be obtained:[R_(x)(π/2){circle around (x)}R_(x)(π/2)][R_(y)(π){circle around (x)}I]:|Ψ⁺>→−|Φ⁻>, |Ψ⁻>→−i|Ψ⁺>|Φ⁺>→−|Ψ⁻>, |Φ⁻>→−i|Φ⁺>  (4.10)

FIG. 11 is a table showing a random permutation of the Bell-basisvectors {|Φ^(±)>} and {|Ψ^(±)>}. In FIG. 11, the phase factor isomitted. The six kinds of transformations shown in FIG. 11 permute thetwo sets of vectors {|Φ^(±)>} and {|Ψ^(±)>} into arbitrary combinations.More specifically, a random integer k (kε{1, . . . , 6}) is picked outand the k^(th) transformation in the table of FIG. 11 is applied to |Σ>.Each of the transformations shown in FIG. 11 is a combination ofsingle-qubit unitary transformations, and can be achieved by beamsplitters. Then, the Bell measurement is performed by the quantumcircuit shown in FIG. 8.

In the present embodiment, the case in which a positron and an electronare used as two particles is described. However, any kinds of particlesmay be used as long as one of them absorbs the other when they approacheach other.

Fourth Embodiment

In a fourth embodiment, a second example of an apparatus for performingthe Bell measurement will be described. First, the Bell states of anelectron-positron pair are defined as follows:

$\begin{matrix}\begin{matrix}{{\left. \Phi^{\pm} \right\rangle = {\left( {1/\sqrt{2}} \right)\left( {{\left. \overset{\_}{0} \right\rangle_{+}\left. \overset{\_}{0} \right\rangle_{-}} \pm {\left. \overset{\_}{1} \right\rangle_{+}\left. \overset{\_}{1} \right\rangle_{-}}} \right)}},} \\{{\left. \Psi^{\pm} \right\rangle = {\left( {1/\sqrt{2}} \right)\left( {{\left. \overset{\_}{0} \right\rangle_{+}\left. \overset{\_}{1} \right\rangle_{-}} \pm {\left. \overset{\_}{1} \right\rangle_{+}\left. \overset{\_}{0} \right\rangle_{-}}} \right)}},} \\{{{{where}\mspace{11mu}\left. \overset{\_}{0} \right\rangle_{+}} = {\;}{{\left. 0 \right\rangle_{a}{\left. 1 \right\rangle_{b^{\prime}}}_{-}\left. \overset{\_}{1} \right\rangle_{+}} = {{\left. 1 \right\rangle_{a}\left. 0 \right\rangle_{b^{\prime}}\left. \overset{\_}{0} \right\rangle_{-}} = {\left. 0 \right\rangle_{c}\left. 1 \right\rangle_{d^{\prime}}}}}}\mspace{14mu}} \\{{{and}\mspace{14mu}\left. \overset{\_}{1} \right\rangle_{-}} = {\left. 1 \right\rangle_{c}{\left. 0 \right\rangle_{d}.}}}\end{matrix} & (5.1)\end{matrix}$The positron travels through paths a and b and the electron travelsthrough path c and d. FIG. 12 shows a quantum circuit for distinguishingthe Bell bases {|Φ^(±)>,|Ψ^(±)>} from one another.

The operation of the quantum circuit shown in FIG. 12 will be describedbelow. One of {|Φ^(±)>,|Ψ^(±)>} is input from the paths a, b, c, and dat t=0. When the basis is | 0>₊| 0>⁻, | 0>₊| 1>⁻, | 1>₊| 0>⁻, or | 1>₊|1>⁻ at t=0, it is transformed as shown in FIG. 13 at t=T₁ and t=T₂. Notethat the paths D and E are replaced with each other between t=T₁ andt=T₂. The paths E and F are observed at t=T₂. If the electron is foundin the path E, it is determined that the input state is |Ψ^(±)>, and ifthe electron is found in the path F, it is determined that the inputstate is |Φ^(±)>. The paths C and D are ignored since the electron isnever present in the paths C and D and the states thereof are always|0>_(C)|0>_(D).

If the electron is found in the path E, |Ψ^(±)> is projected into thefollowing state:(1/√{square root over (2)})(−|0>_(A)|1>_(B)±|1>_(A)|0>_(B)).  (5.2)The operation of the beam splitter H is defined by Expression (2.1), andExpression (5.2) is converted by the beam splitter H as follows:

$\begin{matrix}{\left. {{\left( {1/\sqrt{2}} \right)\left( {{- \left. 0 \right\rangle_{A}}❘1} \right\rangle_{B}} \pm {\left. 1 \right\rangle_{A}\left. 0 \right\rangle_{B}}} \right)\overset{H}{\rightarrow}\left\{ \begin{matrix}{{- \left. 1 \right\rangle_{A}}\left. 0 \right\rangle_{B}} & {{for}\mspace{14mu}\left. \Psi^{+} \right\rangle} \\{{- \left. 0 \right\rangle_{A}}\left. 1 \right\rangle_{B}} & {{for}\mspace{14mu}\left. \Psi^{-} \right\rangle}\end{matrix} \right.} & (5.3)\end{matrix}$Accordingly, the input state is |Ψ⁺> if the positron is found in thepath A at t=T₃, and is |Y⁻> if the positron is found in the path B att=T₃.

If the electron is found in the path F, |Φ^(±)> is projected into thefollowing state:(1/√{square root over (2)})(|0>_(A)|1>_(B)∓|1>_(A)|0>_(B)).  (5.4)Expression (5.4) is converted by the beam splitter H as follows:

$\begin{matrix}{{\left( {1/\sqrt{2}} \right)\left( {{\left. 0 \right\rangle_{A}\left. 1 \right\rangle_{B}} \mp {\left. 1 \right\rangle_{A}\left. 0 \right\rangle_{B}}} \right)}\overset{H}{\rightarrow}\left\{ {\begin{matrix}{\left. 1 \right\rangle_{A}\left. 0 \right\rangle_{B}} & {{for}\mspace{14mu}\left. \Phi^{+} \right\rangle} \\{\left. 0 \right\rangle_{A}\left. 1 \right\rangle_{B}} & {{for}\mspace{14mu}\left. \Phi^{-} \right\rangle}\end{matrix}.} \right.} & (5.5)\end{matrix}$Accordingly, the input state is |Φ⁺> if the positron is found in thepath A at t=T₃, and is |Φ⁻> if the positron is found in the path B att=T₃. Thus, the Bell bases {|Φ^(±)>, |Ψ^(±)>} are distinguished from oneanother.

In the present embodiment, the case in which a positron and an electronare used as two particles is described. However, any kinds of particlesmay be used as long as one of them absorbs the other when they approacheach other.

Fifth Embodiment

In a fifth embodiment, an apparatus for implementing a controlled-NOTgate transformation will be described.

An apparatus for implementing a controlled-NOT gate transformationaccording to the present embodiment uses the method proposed byGottesman and Chuang and implements the controlled-NOT gate indirectlyusing the above-described Bell-state measurement process. Accordingly,the universal set of gates for quantum computation can be realized by aninteraction-free measurement. This means that a desired quantumcomputation algorithm can be implemented by the interaction-freemeasurement.

Gottesman and Chuang have proved that the controlled-NOT gate can beimplemented by generating a four-qubit entangled state expressed as:|χ>=(½)[(|00>+|11>)|00>+(|01>+|10>)|11>]  (6.1)and performing the Bell measurement twice and single-qubit gateoperations (D. Gottesman and I. L. Chuang, “Demonstrating the viabilityof universal quantum computation using teleportation and single-qubitoperations”, Nature (London) 402, 390–393 (1999)). First, a method forgenerating the state |χ> using IFM gates will be discussed.

First, a GHZ state is generated by the quantum circuit shown in FIG. 5,and is input to three pairs of paths (three qubits) of a quantum circuitshown in FIG. 14. The quantum circuit shown in FIG. 14 is used togenerate the state |χ>. Next, each of the three qubits is subjected tothe Hadamard transformation using a beam splitter H. The operation ofthe beam splitters H is expressed as | 0>_(±)→(1/√{square root over(2)})(| 0>_(±)+| 1>_(±)) and | 1>_(±)→(1/√{square root over (2)})(|0>_(±)−| 1>_(±)), and accordingly the following expression is obtained:(1/√{square root over (2)})(| 0>₊| 0>⁻| 0>₊+| 1>₊| 1>⁻| 1>₊)

-   H{circle around (x)}H{circle around (x)}H:

$\begin{matrix}\left. {{->{\left( {1/2} \right)\left\lbrack {\left( {❘{{\overset{\_}{0} >_{+}}❘{{\overset{\_}{0} >_{-} +}❘{\overset{\_}{1} >_{+}}}}} \right)❘{\overset{\_}{1} >_{-}}} \right)}}❘{{\overset{\_}{0} >_{+}{+ \left( {❘{{\overset{\_}{0} >_{+}}❘{{\overset{\_}{1} >_{-} +}❘{{\overset{\_}{1} >_{+}}❘{\overset{\_}{0} >_{-}}}}}} \right)}}❘{\overset{\_}{1} >_{+}}}} \right\rbrack & (6.2)\end{matrix}$Then, | 0>⁻ is added as a fourth qubit, and the third and the fourthqubits are input to an IFM gate. As is understood from FIG. 9, when thecontrol qubit is a positron and the target qubit is an electron, the IFMgate performs the transformation expressed as | 0>₊| 0>⁻→| 0>₊| 0>⁻ and| 1>₊| 0>⁻→| 1>₊| 1>⁻, and accordingly the following expression isobtained:

$\left( {{1/{2\left\lbrack {{\left( {{\left. \overset{\_}{0} \right\rangle_{+}\left. \overset{\_}{0} \right\rangle_{-}} + {\left. \overset{\_}{1} \right\rangle_{+}\left. \overset{\_}{1} \right\rangle_{-}}} \right)\mspace{11mu}\left. \overset{\_}{0} \right\rangle_{+}} + {\left( {{\left. \overset{\_}{0} \right\rangle_{+}\left. \overset{\_}{1} \right\rangle_{-}} + {\left. \overset{\_}{1} \right\rangle_{+}\left. \overset{\_}{0} \right\rangle_{-}}} \right)\mspace{11mu}\left. \overset{\_}{1} \right\rangle_{+}}} \right\rbrack}}\mspace{11mu}\left. \overset{\_}{0} \right\rangle_{-}} \right.$

-   IFM gate:

$\begin{matrix}\begin{matrix}{->{\left( {1/2} \right)\;\left\lbrack {{\left( {{\left. \overset{\_}{0} \right\rangle_{+}\left. \overset{\_}{0} \right\rangle_{-}} + {\left. \overset{\_}{1} \right\rangle_{+}\left. \overset{\_}{1} \right\rangle_{-}}} \right)\mspace{11mu}\left. \overset{\_}{0} \right\rangle_{+}\left. \overset{\_}{0} \right\rangle_{-}} +} \right.}} \\\left. \mspace{34mu}{\left( {{\left. \overset{\_}{0} \right\rangle_{+}\left. \overset{\_}{1} \right\rangle_{-}} + {\left. \overset{\_}{1} \right\rangle_{+}\left. \overset{\_}{0} \right\rangle_{-}}} \right)\mspace{11mu}\left. \overset{\_}{1} \right\rangle_{+}\left. \overset{\_}{1} \right\rangle_{-}} \right\rbrack\end{matrix} & (6.3)\end{matrix}$Thus, if an ideal IFM gate (fidelity=1) can be obtained, the state |χ>is generated with probability 1.

A method for implementing the controlled-NOT gate according to Gottesmanand Chuang is shown in FIG. 15. The thin lines show the paths of qubits,and the bold lines show the paths of classical bits. Reference symbolsB₁ and B₂ denote the Bell measurement. The output of B_(i)(iε{1,2}) isrepresented by two classical bits. More specifically, |Φ⁺>, |Φ⁻>, |Ψ⁺>,and |Ψ⁻> are represented by (x_(i), z_(i))=(0,0), (0,1), (1,0), and(1,1), respectively. σ_(x) is operated when x_(i)=1, and σ_(z) isoperated when z_(i)=1. In addition, no operation is performed whenx_(i)=0 and z_(i)=0. The single-qubit unitary transformations such asσ_(x) and σ_(z) can be achieved by beam splitters.

The Bell measurement in FIG. 15 may be performed by either of the methodaccording to the third embodiment and that of the fourth embodiment.

First, the case in which the Bell measurement in FIG. 15 is performed bythe method according to the third embodiment will be described. In thiscase, the maximum fidelity of a single Bell measurement is ¾. Since theBell measurement is performed twice in FIG. 15, the maximum fidelity ofthe controlled-NOT gate is (¾)²= 9/16>½.

Next, the case in which the Bell measurement in FIG. 15 is performed bythe method according to the fourth embodiment will be described. In thiscase, the maximum fidelity of a single Bell measurement is 1.Accordingly, the maximum fidelity of the controlled-NOT gate is also 1.

From the above discussions, it is understood that the controlled-NOTgate between an arbitrary state of a positron ∀|ψ>₊ and an arbitrarystate of an electron ∀|φ>⁻ can be implemented using IFM gates and beamsplitters. Next, a method for implementing the controlled-NOT gatebetween two states of positrons ∀|ψ>₊ and ∀|φ>₊ is considered. In thiscase, a technique shown in FIG. 16 is used. More specifically, when thestate of a positron |φ>₊ and the state of an auxiliary system | 0>⁻ areprocessed by the controlled-NOT gate twice, as shown in FIG. 16, thewave functions are exchanged and | 0>₊ and |φ>⁻ are obtained.Accordingly, |ψ>₊ and |φ>⁻ can be input to the controlled-NOT gate.

In the present embodiment, the case in which a positron and an electronare used as two particles is described. However, any kinds of particlesmay be used as long as one of them absorbs the other when they approacheach other.

Sixth Embodiment

In a sixth embodiment, apparatuses for generating the Bell state,performing the Bell measurement, and implementing the controlled-NOTgate transformation using an electron and a positron as two particleswill be described.

In first, third, fourth and fifth embodiments, the interaction-freemeasurement using an electron and a positron is considered. Generally,the electron and the positron are generated using an accelerator.Therefore, in the apparatuses for generating the Bell state, performingthe Bell measurement, and implementing the controlled-NOT gate, theinterferometer according to Kwiat et al., the beam splitters, and thepaths through which the electron and the positron travel are containedin a vacuum vessel.

FIG. 17 is a diagram of a system corresponding to the quantum circuitfor generating the Bell state shown in FIG. 1, which is contained in avacuum vessel. The interferometer according to Kwiat et al. shown inFIG. 17 includes five beam splitters. The speeds and paths of theelectron and the positron are adjusted such that the electron and thepositron come near enough to each other at positions shown by thecircles.

Mirrors and beam splitters included in the interferometer may becomposed of metal or insulating plates which function as potentialbarriers for the electron and the positron, metal electrodes with asuitable potential, etc. The reflectance, the transmittance, and thephase shift of the beam splitters are adjusted by the potentialbarriers.

Seventh Embodiment

In a seventh embodiment, apparatuses for generating the Bell state,performing the Bell measurement, and implementing the controlled-NOTgate transformation using an electron and a hole in a semiconductor willbe described.

In first, third, fourth and fifth embodiments, the interaction-freemeasurement using an electron and a positron is considered. However, anykinds of particles may be used as long as one of them absorbs the otherwhen they approach each other. For example, when a conducting electronand a hole in a semiconductor approach each other, pair annihilationoccurs and a photon is generated. Accordingly, the Bell measurement andthe controlled-NOT gate may also be realized by an interaction-freemeasurement using the electron and the hole in the semiconductor. Insuch a case, the beam splitters may be composed of insulating layers orthe like which function as suitable potential barriers for the electronor the hole, and the reflective mirrors may be obtained by controllingpotentials using metal electrodes. The reflectance, the transmittance,and the phase shift of the beam splitters are adjusted by the potentialbarriers.

Eighth Embodiment

In an eighth embodiment, an approximate formula used for evaluating thefidelity of an IFM gate when the absorption probability of an object isless than 1 will be described.

The above-described IFM gate includes the beam splitters and uses theinteraction between a photon and an absorbing object. The beam splittersare commonly used as experimental components, and those with highprecision are available. In comparison, the absorbing object cannot beexpected to absorb the photon with probability 1 when the photonapproaches the absorbing object. Accordingly, in the present embodiment,the responsibility of the IFM gate is evaluated on the assumption thatthe absorbing object absorbs the photon with probability (1−η) and failsto absorb the photon with probability η when the photon approaches theabsorbing object.

Accordingly, in the interferometer shown in FIG. 22, the state in whichthe photon comes out from the beam splitters into the path a, that is,the state | 0>=|1>_(a)|0>_(b), is transformed as follows:| 0>→√{square root over (η)}| 0>+√{square root over (1−η)}|absorption>,0<η<1  (7.1)where |absorption> is the state in which the photon is absorbed by theobject. |absorption> is orthogonal to {| 0>, | 1>} and is standardized,where | 0>=|1>_(a)|0>_(b) and | 1>=|0>_(a)|1>_(b). When anelectron-positron pair is used, |absorption>corresponds to the state inwhich the photon γ is generated. In addition, when a photon-atomreaction is used, |absorption> means that the atom is in the state |e₂>.

For convenience, the transformation of the photon is expressed by amatrix with the basis {| 0>, | 1>} as follows:

$\begin{matrix}\begin{matrix}{{\left. \overset{\_}{0} \right\rangle = {{\left. 1 \right\rangle_{a}\left. 0 \right\rangle_{b}} = \begin{pmatrix}1 \\0\end{pmatrix}}},} & {\left. \overset{\_}{1} \right\rangle = {{\left. 0 \right\rangle_{a}\left. 1 \right\rangle_{b}} = \begin{pmatrix}0 \\1\end{pmatrix}}}\end{matrix} & (7.2)\end{matrix}$Accordingly, Expression (1.3), which defines the operation of the beamsplitter B, is rewritten as follows:

$\begin{matrix}\begin{matrix}{{B = \begin{pmatrix}{{\cos\mspace{11mu}\theta}\;} & {\sin\mspace{11mu}\theta} \\{{- \sin}\mspace{11mu}\theta} & {\cos\mspace{11mu}\theta}\end{pmatrix}},} & {\theta = {{\pi/2}N}}\end{matrix} & (7.3)\end{matrix}$In addition, Expression (7.1), which shows the absorption process of thephoton, is also rewritten as follows:

$\begin{matrix}\begin{matrix}{{A = \begin{pmatrix}\sqrt{\eta} & 0 \\0 & 1\end{pmatrix}},} & {0 < \eta < 1}\end{matrix} & (7.4)\end{matrix}$Since Expression (7.1) involves the dissipation (annihilation) of thephoton, A is not unitary. The probability that the photon input as |1>=|0>_(a)|1>_(b) will be detected as | 1>=|0>_(a)|1>_(b) after passingthrough the N beam splitters is calculated as follows:P=|> 1|(BA)^(N−1) B| 1>|²  (7.5)The IFM gate operates correctly at least with probability P. FIG. 18 isa graph showing lines obtained by connecting the results of numericalcalculations of the probability P as a function of N and η usingExpressions (7.3), (7.4), and (7.5). More specifically, FIG. 18 is agraph of the fidelity of the IFM gate plotted as a function of thenumber N of beam splitters (the number of times the particle B hits thebeam splitters) and the probability η that the absorbing object(particle A) will fail of absorb the photon (particle B). The bold linesshow the results of exact numerical calculations, and the dashed linesshow the results obtained by the approximate formula satisfied when N islarge. The four bold lines correspond to η=0, 0.05, 0.1, and 0.2 fromthe top.

In order to evaluate the influence of the noise η on the responsibilityof the IFM gate, the change which occurs in the probability P when η isfixed to a finite value and N is increased to infinity (N→∞) will bediscussed below. The evaluation of the probability P obtained byExpression (7.5) under the condition N→∞ is difficult in that thedependence of P on N comes from θ=π/2N in the matrix B in Expression(7.3) and the exponent N of (BA)^(N−1) in Expression (7.5).

With regard to θ=π/2N in the matrix B, the expression may be expanded inpowers of θ and high-order items may be ignored. Accordingly, inExpressions (7.3), (7.4), and (7.5), η is fixed to a value in the rangeof 0<η<1 and the powers of θ up to the second order are calculated underthe condition θ=π/2N→0. First,

$\begin{matrix}{B = {\begin{pmatrix}{1 - \left( {\theta^{2}/2} \right)} & \theta \\{- \theta} & {1 - \left( {\theta^{2}/2} \right)}\end{pmatrix} + {O\left( \theta^{3} \right)}}} & (7.6)\end{matrix}$is obtained, and

$\begin{matrix}\begin{matrix}{({BA})^{k} =} \\{\begin{pmatrix}{{\sqrt{\eta}}^{k} - {\theta^{2}\begin{bmatrix}{{\sum\limits_{l = 1}^{k - 1}{l{\sqrt{\eta}}^{l}}} +} \\{\left( {k/2} \right){\sqrt{\eta}}^{k}}\end{bmatrix}}} & {\theta\;{\sum\limits_{l = 0}^{k - 1}{\sqrt{\eta}}^{l}}} \\{{- \theta}\;{\sum\limits_{l = 0}^{k - 1}{\sqrt{\eta}}^{l + 1}}} & {1 - {\theta^{2}\begin{bmatrix}{{\sum\limits_{l = 1}^{k - 1}{l{\sqrt{\eta}}^{k - l}}} +} \\\left( {k/2} \right)\end{bmatrix}}}\end{pmatrix} + {O\left( \theta^{3} \right)}}\end{matrix} & (7.7)\end{matrix}$where k=1, 2, . . . , is obtained by induction

$\left( {\sum\limits_{l = 1}^{0}\mspace{14mu}{{means}\mspace{14mu}{that}\mspace{14mu}{the}\mspace{14mu}{sum}\mspace{14mu}{is}\mspace{14mu}{not}\mspace{14mu}{calculated}}} \right).$means that the sum is not calculated). Accordingly, the amplitude of thephoton at the path | 1>=|0>_(a)|1>_(b) after passing through the N beamsplitters is expressed as follows:

$\begin{matrix}\begin{matrix}{{\left\langle \overset{\_}{1} \right.\mspace{11mu}\left( {B\; A} \right)^{N - 1}B\;\left. \overset{\_}{1} \right\rangle} =} \\{{1 - {\theta^{2}\left\lbrack {\left( {N/2} \right) + {\sum\limits_{l = 0}^{N - 2}{\sqrt{\eta}}^{l + 1}} + {\sum\limits_{l = 1}^{N - 2}{l{\sqrt{\eta}}^{N - 1 - l}}}} \right\rbrack} + {O\left( \theta^{3} \right)}} =} \\{{1 - {\theta^{2}\left\lbrack {\left( {N/2} \right) + {N\;{\sum\limits_{l = 1}^{N - 1}{\sqrt{\eta}}^{l}}} - {\sum\limits_{l = 1}^{N - 1}{l{\sqrt{\eta}}^{l}}}} \right\rbrack} + {O\left( \theta^{3} \right)}} =} \\{1 - {\left( {\pi/2} \right)^{2}\;{\left( {1/N} \right)\;\left\lbrack {\left( {1/2} \right) + \frac{\sqrt{\eta}\left( {1 - {\sqrt{\eta}}^{N - 1}} \right)}{1 - \sqrt{\eta}} -} \right.}}} \\{\left. {\left( {1/N} \right)\;\frac{\sqrt{\eta}\left\lbrack {1 - {N{\sqrt{\eta}}^{N - 1}} + {\left( {N - 1} \right){\sqrt{\eta}}^{N}}} \right\rbrack}{\left( {1 - \sqrt{\eta}} \right)^{2}}} \right\rbrack + {O\left( \theta^{3} \right)}}\end{matrix} & (7.8)\end{matrix}$where N=2, 3, . . . In the above transformation, the following formulasare applied:

$\begin{matrix}{{\sum\limits_{k = 1}^{N}x^{k}} = {{x\left( {1 - x^{N}} \right)}/\left( {1 - x} \right)}} & (7.9) \\{{\sum\limits_{k = 1}^{N}{k\; x^{k}}} = {{x\left\lbrack {1 - {\left( {N + 1} \right)\; x^{N}} + {N\; x^{N + 1}}} \right\rbrack}/\left( {1 - x} \right)^{2}}} & (7.10)\end{matrix}$Expression (7.8) is regarded as the approximate formula of >1|(BA)^(N−1)B| 1>, where N is sufficiently large. Note that O(θ³) inExpression (7.8) may include a (1/N)^(th)-order term. Accordingly,Expression (7.8) cannot be regarded as an approximate formula up to the(1/N)^(th) order.

In FIG. 18, the dashed lines are obtained by connecting the resultsobtained by the approximate formula of P=|> 1|(BA)^(N−1)B| 1>|²calculated using Equation (7.8) as a function of N and η (the fourdashed lines correspond to η=0, 0.05, 0.1, and 0.2 from the top). As isclear from FIG. 18, the results of approximation of P obtained usingExpression (7.8) is close to those obtained by the exact numericalcalculations when N is large.

As is understood from FIG. 18, for any of η=0, 0.05, 0.1, and 0.2, P=|>1|(BA)^(N−1)B| 1>|² approaches 1 as N increases. This means that ifthere is no limit to the number of beam splitters, the influence ofnoise calculated by Expression (7.1) can be eliminated. When thetransmittance of the beam splitters is set sufficiently low (when N issufficiently large and θ is sufficiently small), the probability thatthe photon will approach the absorbing object reduces, and accordinglythe influence of the probability η that the object will fail to absorbthe photon also reduces. When N is extremely large, the transmittanceT=sin²(π/2N) of the beam splitters is set extremely low, which isequivalent to the case where beam splitters with T=sin²(π/2N)˜(1/N²) areprovided. This can be interpreted as the noise in the interactionbetween the photon and the object being compensated for by the precisionof the beam splitter.

Next, the number N of beam splitters required for increasing thefidelity of the IFM gate to a predetermined value P(0<∀P<1) will bedetermined as a function of η. If N suddenly increases as η starts toincrease from 0, the anti-noise performance of the IFM gate isrelatively low.

An approximate formula of Expression (7.8) is used for the aboveevaluation. In Expression (7.8), it is assumed that η is relatively lowand N√{square root over (η)}^(N)<<1 is satisfied when N is sufficientlylarge. In this case, P can be simply expressed as follows:

$\begin{matrix}{{\left. P \right.\sim 1} - {\left( {{\pi^{2}/2}N} \right)\left\lbrack {\left( {1/2} \right) + \frac{\sqrt{\eta}}{1 - \sqrt{\eta}}} \right\rbrack}} & (7.11) \\{{Accordingly},} & \; \\{{\left. N \right.\sim{\left( {1/2} \right)\left\lbrack {\pi^{2}/\left( {1 - P} \right)} \right\rbrack}}\frac{1 + \sqrt{\eta}}{1 - \sqrt{\eta}}\mspace{14mu}{or}} & (7.12) \\{{\log{\left. \frac{1 + \sqrt{\eta}}{1 - \sqrt{\eta}} \right.\sim\log}\; N} + {Const}} & (7.13)\end{matrix}$where Const=log [2(1−P) /π²] is obtained. Expression (7.13) is satisfiedwhen P is sufficiently close to 1, that is, when N is sufficientlylarge. A graph of Expression (7.13) is shown in FIG. 19, in which thenumber N of beam splitters (the number of times the particle B hits thebeam splitter) required for increasing the fidelity of the IFM gate tothe predetermined value is plotted as a function of the probability ηthat the absorbing object (particle A) will fail to absorb the photon(particle B). The vertical axis of the graph is drawn with a logarithmicscale. In Expressions (7.11) and (7.13), N suddenly increases as ηincreases, and diverges to infinity (N→∞) when η→1. For example, Nobtained when η=¼ is three times of that obtained when η=0.

Although the present invention has been described in its preferred formwith a certain degree of particularity, many apparently widely differentembodiments of the invention can be made without departing from thespirit and the scope thereof. It is to be understood that the inventionis not limited to the specific embodiments thereof except as defined inthe appended claims.

1. An apparatus for generating a quantum state of a two-qubit systemincluding two qubits, each qubit being represented by a detectorprobability, where the probability is based upon N samples of a systemin which a particle invariably travels through one of two paths, theapparatus comprising: a first beam splitter configured to receive afirst particle of two particles having no correlation with each other,and to output the first particle into two paths in a superposition; andan interferometer configured to implement an interaction-freemeasurement, wherein an interaction-free measurement is a probabilitymeasurement method defined as: the condition when, after the N samplesof the system containing the two particles, the probability of detectionof a particle after N samples by a detector is about P_(C)=cos² θ sin²θ, wherein N is defined as N=1/(1−P_(B))=1/(1−cos⁴ θ), wherein θ is acharacteristic of the first beam splitter, wherein P=cos⁴θ, and whereinthe interferometer is configured to receive a second particle of the twoparticles and the output of the first beam splitter and generates a Bellstate with asymptotic probability
 1. 2. An apparatus according to claim1, wherein the interferometer includes a cavity and second beamsplitters sectioning the cavity into two chambers, wherein the twoparticles are input into different chambers of the cavity, the firstparticle absorbing the second particle if the first particle and thesecond particle come near enough to each other, and wherein the secondparticle successively hits the second beam splitters so that thetransmitted wave component in the wave function of the second particletravels back and forth between the two chambers.
 3. An apparatusaccording to claim 2, wherein the particle transmittance of the secondbeam splitters is set to a predetermined value or less so that theprobability amplitude of the state in which the second particle isabsorbed by the first particle by entering the chamber containing thefirst particle when the second particle hits each of the beam splitteris set small, and wherein the first and the second particles repeatedlyapproach each other with an extremely small probability amplitude sothat the first particle absorbs the second particle with probabilityclose to zero, whereby the second particle is put into differentchambers depending on whether the first particle is input to the cavity.4. An apparatus according to claim 3, wherein the first beam splitteroutputs the first particle to one of the chambers while the firstparticle is in a quantum superposition of present and absent states, sothat the first particle and the second particle are put into the Bellstate with asymptotic probability
 1. 5. An apparatus according to claim3, wherein the first beam splitter implements a Hadamard transformationand thereby inputs the first particle to one of the chambers while thefirst particle is in a quantum superposition of present and absentstates, so that the Bell state is generated with asymptotic probability1 if the number of times the second particle hits the beam splitters inthe interferometer is large.
 6. An apparatus according to claim 3,wherein the two particles are photons and the Bell state is generatedusing an auxiliary system including a three-level atom by regarding aground state in which the atom can absorb the photons as a state inwhich the second particle is absorbed by the first particle and a firstexcited state in which the atom cannot absorb the photons as a state inwhich the second particle is not absorbed by the first particle.
 7. Anapparatus according to claim 6, wherein a transition of the atom betweenthe ground state and the first excited state is implemented by Rabioscillation, and the energy of the two photons is the same as thedifference in energy level between the ground state and a second excitedstate of the atom.
 8. An apparatus according to claim 3, wherein the twoparticles are a positron and an electron and the Bell state is generatedby regarding a state in which a photon is generated by pair annihilationof the positron and the electron as the state in which the secondparticle is absorbed by the first particle, the pair annihilationoccurring if the positron and the electron come near enough to eachother.
 9. An apparatus according to claim 3, wherein the two particlesare a hole in a semiconductor and a conducting electron and the Bellstate is generated by regarding a state in which a photon is generatedby pair annihilation of the hole and the conducting electron as thestate in which the second particle is absorbed by the first particle,the pair annihilation occurring if the hole and the conducting electroncome near enough to each other.